数学图形之Breather surface

这是一种挺漂亮的曲面图形,可惜没有找到太多的相关解释.

In differential equations, a breather surface is a mathematical surface relating to breathers.

其数学公式很复杂,参数方程为:

\begin{align} x & {} = -u+\frac{2\left(1-a^2\right)\cosh(au)\sinh(au)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)} \\ \\ y & {} = \frac{2\sqrt{1-a^2}\cosh(au)\left(-\sqrt{1-a^2}\cos(v)\cos\left(\sqrt{1-a^2}v\right)-\sin(v)\sin\left(\sqrt{1-a^2}v\right)\right)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)} \\ \\ z & {} = \frac{2\sqrt{1-a^2}\cosh(au)\left(-\sqrt{1-a^2}\sin(v)\cos\left(\sqrt{1-a^2}v\right)+\cos(v)\sin\left(\sqrt{1-a^2}v\right)\right)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)} \end{align}

where 0 < a < 1.

维基的相关网址为:http://en.wikipedia.org/wiki/Breather_surface

使用自己定义语法的脚本代码生成数学图形.相关软件参见:数学图形可视化工具,该软件免费开源.QQ交流群: 367752815

#http://xahlee.info/surface/breather_p/breather_p.html

vertices = D1:100 D2:100

u = from -13.2 to 13.2 D1
v = from -37.4 to 37.4 D2

b = 0.4
r = 1 - b*b
w = sqrt(r)

d = b*((w*cosh[b*u])^2 + (b*sin[w*v])^2)

y = -u + (2*r*cosh[b*u]*sinh[b*u])/d
z = (2*w*cosh[b*u]*(-(w*cos[v]*cos[w*v]) - sin[v]*sin[w*v]))/d
x = (2*w*cosh[b*u]*(-(w*sin[v]*cos[w*v]) + cos[v]*sin[w*v]))/d

使用随机数

#http://en.wikipedia.org/wiki/Breather_surface

vertices = D1:100 D2:100

u = from -13.2 to 13.2 D1
v = from -37.4 to 37.4 D2

a = rand2(0.1, 0.9)
w = sqrt(1 - a*a)

d = a*((w*cosh[a*u])^2 + (a*sin[w*v])^2)

y = -u + (2*(1 - a*a)*cosh[a*u]*sinh[a*u])/d
z = (2*w*cosh[a*u]*(-(w*cos[v]*cos[w*v]) - sin[v]*sin[w*v]))/d
x = (2*w*cosh[a*u]*(-(w*sin[v]*cos[w*v]) + cos[v]*sin[w*v]))/d

posted on 2014-08-22 11:11 叶飞影阅读(1851) 评论(0) 编辑收藏举报